Collaborative learning and the construction of common knowledge

European Journal rf Psychology of Education2000, Vol, XV, ,," 4, 479-490(i) 2000, I.S.P.A.

Collaborative learning and the construction ofcommon knowledge

Ed ElbersLeen Streefland tUtrecht University, The Netherlands

In collaborative learning, students not only have to acquireknowledge, they also have to learn to regulate the process of acquiringknowledge. In the traditional classroom, the function of regulation restswith the teacher. In the collaborative classroom, however, theresponsibility for learning has in part been handed over to the students.We examine how students, who work as members of a community oflearners, construct shared understanding. In particular, we want toexplore what interactive and discursive tools students use in theircollaboration. We present observations made during a series ofinnovative mathematics lessons in an 8th grade classroom at a Dutchprimary school in which children (between I I and I3 years of age)worked as "researchers" who were encouraged to formulate questionsfor exploration and to collaborate in answering them. Both in smallgroup discussions and in discussions involving the whole class,students worked on the construction of arguments and the creation ofshared knowledge. The construction and diffusion of knowledgeoccurred in "cycles of argumentation" to which many childrencontributed and in which ideas were repeated and elaborated upon.Because, in students' collaboration, learning is made dependent onproposing and critically discussing arguments, the character ofknowledge, acquired under these circumstances, is different fromknowledge acquired in a more traditional classroom setting,

Introduction

This contribution examines the development of common knowledge in the collaborativeclassroom. Despite the attention given to collaborative learning in recent educational research,few studies have examined how knowledge comes to be shared in classrooms where studentslearn in collaborating groups (Hatano & Inagaki, 1991). Researchers have typically focused onargumentational aspects of discourse and on ways of motivating and guiding pupils tocontribute to discussions and to weigh arguments critically (Mercer, 1995; Forman,Larrcarncndy-Joems, Stein, & Brown, 1998; Rojas-Drummond, Hernandez, Velez, &Villagran, 1(98). However, students not only have to participate in discussions and present

480 E. EWERS & L. STREEFLAND

and evaluate ideas, they also have to present their contributions in such a way that they can beshared by their fellow students. Students' talk in a collaborative classroom is not only forexpressing and discussing ideas. It also has to provide ways for sharing insights and reachingjoint understanding. This is an important aspect to collaborative learning on which we willconcentrate in this article.

In classrooms, pupils, guided by their teacher, build an expanding base of sharedunderstandings, which Edwards and Mercer (1987) have called "common knowledge". Theseauthors view education as a communicative process and demonstrate how joint understandingin the classroom is the product of the presentation and reception of knowledge, of negotiationsand discussions following misunderstandings. Their research in British junior schools showsin some detail how teachers use particular didactic tools for bringing about commonknowledge.

One important tool used by teachers in Edwards and Mercer's study was to ask questions.Most of the talk in classrooms observed in their study emanated from the teacher and much ofthat talk was in question form. Most utterances by children fell within the category of elicitedcontributions. They were part of the IRF pattern of teacher-pupil interaction: an Invitation bythe teacher (mostly a pseudo-question) was followed by a Response by the pupil, to which theteacher reacted with Feedback (for a discussion of the IRF pattern, see Wells, 1993; andCandela, 1999). Another tool available to the teacher was recapitulating part of the subjectmatter, not by repeating it in a literal way but by applying it to new situations or linking it to amore general context. Edwards and Mercer's analyses show that such recapitulations oftenform a significant part of the feedback statements which teachers make as the last part of theIRF pattern.

Teachers use the IRF pattern and recapitulation statements for creating commonknowledge in the classroom. By asking a question, the teacher focuses the pupil's attention oncertain issues or experiences and creates an opportunity for checking whether that pupil hasacquired the necessary knowledge. By doing so, the teacher calls up previous understandingand constructs a shared mental context for the presentation of new information. In thefeedback statement, the teacher tells the pupil if the answer was correct. At the same time theteacher, by recapitulating the answer, shows how it can be formulated or conceptualized in amore general or sophisticated manner. The recapitulation allows the teacher to expand thepupil's understanding or to introduce new knowledge. The previous evocation of alreadyexisting insights functions as a scaffold for making it easier for the student to appropriate thenew information. With the discursive tools of IRF and recapitulation, the teacher createscontinuity in the discourse by connecting new knowledge to previous understanding.

Since they depend on the teacher's control of classroom talk, the discursive tools of IRFand recapitulation, are not suitable for characterizing the organization of learning in aclassroom in which children work collaboratively (Elbers, Derks, & Streefland, 1995; Formanet al., 1998; Candela, 1999). For their collaboration, students have to apply discursive patternswhich allow them to work without being dependent, in every instance, on the teacher'sinitiative or feedback. These patterns used by the students during their collaboration must fitthe demands of establishing common knowledge. One demand for the participants in acollaborative classroom is to create a continuous discourse which gives sufficient opportunityfor many pupils to be involved and to contribute to the construction of understanding. Anotheris to find interactive tools for the distribution and sharing of knowledge,

Our focus will be on collaborative learning in classrooms in which pupils have beenencouraged to use their initiative and to work together at pursuing their intellectual interests.Recent innovations of collaborative learning have created learning communities in theclassroom (Seixas, 1993; Brown & Campione, 1994; Rogoff, 1994; Scardarnalia & Bereiter,t996; Elbers & Strcetland, 2(00). These projects are based on the understanding that children'slearning is not so much an individual, but a communal activity. Children construct meaningsand insights as members of a "community of learners" or a "community of inquiry", throughinteraction with others, through discussions and negotiations, by proposing, criticizing andrejecting hypotheses. Addressing pupils as members of a community of inquiry means inviting

COLLABORATIVE LEARNING AND COMMON KNOWLEDGE 481

them to actively state questions and to work at solving these by participating in discussions,making investigations and doing experiments (Seixas, 1993).

In a community of inquiry, learning is not based on the authority of the teacher or on theguidelines of a curriculum, but occurs because children contribute to the construction ofknowledge for which they themselves, to a certain extent, have been made responsible.Students learn by participating in collective knowledge building activities, not by preparingthemselves for individual tests. Educational projects based on a community of inquiryapproach, as reported on by Rogoff (1994), Brown and Campione (1994) and Scardamalia andBereiter (1996), have shown that students are able to contribute to the organization of theirlearning. Their co-responsibility for the events in the classroom allows them to express theirinterests in knowing and to discuss the purpose of their activities. The pupils make aconsiderable number of spontaneous contributions which are not responses to questions by theteacher, but which are initiatives by the children in an ongoing dialogue. The teacher providessupport and guidance without controlling all interactions during the lessons. Under thesecircumstances, "new cultures of schooling" (Scardamalia & Bereiter, 1996) arise that placeunderstanding, rather than the reproduction of knowledge, as the main focus. This approachdiffers from the traditional classroom in which the teacher controls the content and course ofany discussior and in which the curriculum or the textbook, rather than students' everydayexperiences, forms the starting point for learning.

How is common knowledge constructed in a community of inquiry? What discursivetools do the pzrticipants in a collaborative classroom have at their disposal for creating sharedcontexts and for connecting new to given insights? How is joint understanding created in aclassroom in which students have been made responsible for their own learning and in whichthe discourse, or important parts of it, is not dominated by the teacher? Let us assume that agood idea has been put forward by one of the pupils in a group of four pupils working together.How do the other three pupils come to share this insight? Next, suppose that all members ofthis group have appropriated the idea and that they have presented it to their fellow students ina whole class discussion, what about the diffusion of this knowledge to these other students?How do we know that the result really becomes common knowledge, that it is shared by otherpupils? In short, how do ideas "migrate" in the classroom and what discursive tools do theparticipants me for making this migration possible?

In order to answer these questions we will look at discussions in a classroom in whichstudents and teachers worked as members of a community of inquiry. Since our studyinvolved both small group discussions and class discussions, it is particularly suited forexamining the migration of ideas throughout the classroom and the negotiations which takeplace in the process of mutual appropriation.

The setting of the study

The setting for our study was an 8th grade classroom at a Dutch primary school (childrenbetween 11 and 13 years of age). This particular class was taught a series of innovativemathematics lessons prepared jointly by the teacher and the Freudenthal Institute forMathematics Education at Utrecht University. The educational aim was to involve children inthe construction of mathematical knowledge by inviting them to work as members of acommunity of inquiry. The experimental lessons spanned a period of four months; there was alesson of about one and a half hours once a week. At the beginning of each lesson, theteachers repeated what it meant to be members of a community of inquiry: we are researchers,let us do research. During the lessons the children were addressed as researchers. The teachers(there were two teachers, the regular teacher and the second author) referred to themselves notas teachers, but as senior researchers.

A typical lesson started with the teachers introducing a broad topic or a general problem(often with reference to some example from everyday life, a newspaper clipping, a photo, etc.)

482 E. ELBERS & L. STREEFLAND

that was to be the subject for the classroom activities of that particular day. Next, the studentswere invited to work on this topic by formulating research questions and by developinganswers to these questions. Work in small groups of 4 or 5 children alternated with classdiscussions in which the results of the groups were made available for discussion in the wholeclass. There was no individual work. The children were encouraged to listen to one another, totake others' ideas seriously and to use arguments for convincing others. As the children in thisclassroom had already completed most of the regular mathematics program for their grade, theteachers had considerable opportunity during the experimental lessons to follow the children'ssuggestions and solutions to problems.

The motivation to set up the experimental series of lessons originated from a view onmathematics learning: Realistic Mathematics Education (Freudenthal, 1991; Goffree, 1993;Streefland, 1993). The activities of teachers and learners did not aim at learning mathematicalsubject matter in the first place, but rather at learning to "mathernatize", that is, turningeveryday issues into mathematical problems and using the mathematics evolving from theseactivities to solve realistic problems. Mathematics, taught and learnt in this way, loses itscharacter as merely school knowledge, because it helps students to link everyday knowledgeand mathematics, or rather, to enrich common sense with mathematical understanding. Theclassroom activities stimulated the students to make their own mathematical constructions andproductions and, in this process, use both their informal understanding of everyday-lifesituations and the mathematical means already available to them.

The observations presented in this article are part of a larger project of observations inthese mathematics lessons which we undertook because we wanted to understand howchildren learn in a community of inquiry and how new forms of cooperation among pupils andbetween pupils and teachers were created (Elbers, Derks, & Streetland, 1995; Elbers &Streefland, 2000; Streefland & Elbers, 1996). In this project we observed several lessons; wemade video and audio recordings of both the class discussions and the discussions in smallgroups. These recordings were transcribed. The conversation sequences discussed in this

article have been chosen from these transcripts and have been translated by us into English. Inorder to improve their readability, we present them in a non-technical way, mostly leavingaside information on pauses, overlapping speech and pronunciation errors (followingMercer's, 1995, example). The teachers are indicated by the abbreviations TI for the class'sregular teacher and T2 for Leen Streefland of the Freudenthal Institute.

Observations

For the purposes of this article: to study the migration of ideas and the establishment of jointunderstanding, we will discuss long sequences from one lesson. The broad task given to thechildren in this lesson was about scale. The school had recently been accommodated in a newbuilding and the regular class teacher had taken the opportunity to start a learning projectincluding drawing a map of their home town with the new school on it and the construction of ascale model of the school. The mathematics lessons supported these activities by discussing theconcept of scale. During the lesson from which the sequences have been taken, the children wereasked to address the problem of estimating the height of buildings. They had recently worked onthe problem of measuring distances in order to be able to draw a map of the school and itssurroundings. Now, at the start of the lesson, the teachers introduced a new problem.

Sequence 1 (class discussion)

Teacher 2: How can we find out the height of the church or a school?

Teacher I: Without asking somebody, by doing research.

Pupils: Making estimates.


T2: Also which instruments could we use? There are at least as many ways as we had forfinding out the distance from home to school.

( ... )T2: So. try to imagine that you have the task of finding out the height of this school or of

St. John's church.

T 1: Or the water tower.

Immediately after Sequence 1, the teachers encouraged the students to work on thisproblem and hinted that one way of estimating the height of a building is by making use ofshadows. However, they also emphasized the variety of possible solutions. The children weretold that they could make drawings to support their solutions. They then started working insmall groups.

Sequence 2 presents the discussion in one of these groups (group 5).

Sequence 2 (discussion in group 5, consisting offive children: Bart, Peter, Remco, Richard,and Roberto)

(1) Richard:

(2) Roberto:

(3) Bart:(4) Roberto:(5) Bart:(6) Remco:

(7) Richard:

(8) Roberto:(9) Remco:

(10) Roberto:

(11) Roberto:

(12) Remco:

(13) Roberto:

(14) Roberto:(15) Remco:

What are we supposed to do?

Make estimates ... the water tower.

You need to look at the sun.The shadow.The shadow is sometimes that long or that short.When the sun is exactly above the tower, the shadow equals the tower.

Of course it isn't. Remco, when it is above there is no shadow at all. It should behere, with light diagonally from above.(making a drawing) Here is the tower and there is the shadow.

If you change it (=the shadow) ...make it a bit longer or shorter, the sun changes position.

When it is precisely above, then there is no shadow.

A very short shadow.Very small. But when the sun is here (indicates on the drawing) then the shadow istwice as long as the tower.So, the sun should be diagonally opposite the tower.

It depends on where the sun is.

After the students have formulated their task (1-4), Bart explains that the shadow of atower changes all the time (5). Then they start working out a model of these changes. Remco'sidea that the shadow equals the tower when the sunlight comes exactly from above (6) isrefuted (7) and this leads to a discussion about the relationship between the position of the sunand the changing shadow (9-13). The pupils make a drawing as support for their discussion. Inthe end, coming to a conclusion (14-15), it seems that Roberto and Remco do not completelyagree as to what they are working out. Roberto's statement refers to a situation in which theshadow equals the tower (14), whereas Remco's conclusion refers to the general model of thechanging shadow (15).

We see in Sequence 2 that children help each other to set the problem and to come to asolution. Part of this work is to criticize statements and to suggest alternative solutions. Theinteresting discussion in Sequence 2 has a repetitive or "cyclical" nature. At first, the idea thatthe shadow equals the tower when the sun is above the tower is merely contradicted (6-7).Later, when the relationship between the position of the sun and the changing shadow hasbeen worked out in a model, the situation in which the shadow is very short is explained withthe help of the model (11-13).


After the group discu ssion in Sequ en ce 2, there was a whole class co nve rsa tion in whichthe groups were asked to present the co ncl us ions of the ir work for ge ne ra l di scussion. Th efo llo wing Sequence is taken from th is d iscussion in the who le class.


(2) Teacher :

( 19) Saskia:

(20) Teacher:

Teacher =TI

( 1) Teacher:

(3) Micha:

(4) Teacher:

(5) Micha:

(6) Patrick:

(7) Demi:

(8) Teacher:

(after a discussion in which the known height of smaller buildings was used toestimate the unkn own height of the tower ) Wh at other ways do we have thanwith the help of a building?

(responding to children volunteering answers) First Micha, after him Rernc oand then Bart,

Use the shadow .

How can yo u use the shadow?

Measure the shadow and then make a calculation of the height of the tower.

About two time s.

No, sir , may I?

(responding to Micha) A very good idea. Here is the tower, here is the shadow,because there is the sun. 1 measure the shadow, twenty meter s, so, conclusion,the tower is twent y meters?

No.

No, tak e it twice .

(addre ssing Micha) Th en something is missing in yo ur argument.

It is longer.

What do you mean, the tower or the shado w'?

The tow er.

Th e shadow .

How do yo u know?

Of co urse, not . It depend s on the position of the sun.

Peter asks: how do you know '? Saskia says : it depend s on where the sun is.Could you expl ain , Saskia?

When the sun is above, then 1 think that the shadow is shorter.

So, when the sun is high, then the shadow is shorter. And when the sun issetting?

(21) Saskia: Then the shadow is longer.

(22) Teacher: Does everybody agree with Saskia?

(23) Several pupils: Yes.

(24) Teacher : So, it is wrong to measure the shadow and say: that is the height of the tower.Something has to be done with the shado w.

(9 ) Micha:

( 10) Patrick:

( I I) Teacher:

(J2) Micha:

(13) Te acher :

(14 ) Mich a:

(15) Monique:

(16) Peter :

(17) Sas kia:

( 18) Tea cher:

This class di scuss ion sta rts with the refutation of the sta teme nt that the len gth of theshadow equals the hei ght o f the towe r or that there is an invariable relat ion ship bet ween shadowand tower (5- 15). Sa sk ia then br ings in the argument that the shadow gets shorter when thesun rise s, whereas it becomes longer when the sun sets . It is interesting that , at the beginningof the di scu ssion , Remco and Bart, two members of group 5 (see Sequen ce 2), volunteer butdo not get a chance to parti cip ate or co ntribute. No doubt they would ha ve co mmunicated theco ncl usions of their gro up to the who le class.


The class discussion leads to an understanding of the relationship between the shadowand the position of the sun, but this understanding is less sophisticated than the model workedout in group S (in Sequence 2). In particular, what is lacking is how the relationship betweenthe height of the tower and the length of the shadow can be formulated with some precision.Group 5 (in Sequence 2) had worked out three positions, the first in which the shadow is twicethe tower's height, the second in which the shadow equals the tower and the third when theshadow is very short or non-existent because the sun is above the tower. Nonetheless, the classdiscussion of Sequence 3 leads to common knowledge, which can now be used for a furtherexploration of the problem. In Sequence 4, which immediately followed the discussiontranscribed in Sequence 3, the results of the discussion in group 5 do play an important role inthe discussion. Bart, a member of group 5, tries to present the conclusion of this group to thewhole class, but initially fails to put forward his argument in a convincing way.


Teacher =TI

(I) Bart:

(2) Teacher:

(3) Bart:

(4) Teacher:

If you think of a clock (points to the clock on the wall in front of the classroom),then nine is west and three is east. If the sun is in the north-east and the north-west,then it is about all right.

You think that when the sun is at a certain height, then you can use the shadow forcalculating the height of the tower.

Yes, it should be diagonally above the tower.

You mean, then the shadow equals the height of the tower.

Nobody except the teacher responds to Bart's idea. Then Peter, also from group 5, makesa suggestion.

(5) Peter:

(6) Teacher:

(7) Peter:

(8) Teacher:

(9) Peter:

(10) Richard:

(11) Teacher:

(12) Micha:

(13) Teacher:

(14) Micha:

(IS) Teacher:

(16) Saskia:

(17) Teacher:

Nail a strip of wood to the tower at a height of two meters. The sun is behind thetower, the strip is on the side of the tower, so you see the shadow of the strip.

Wait a second. I'll make a drawing on the blackboard (he makes a drawing). Hereis the shadow of the tower and here is the shadow of the strip.

Now you can calculate the scale of the shadow.

(pointing to the blackboard) Suppose the sun is very low and the distance of thebase of the tower to the shadow of the strip is six meters.

Then you know what scale has been used and you can calculate the height of thetower.

On that particular time of the day. The tower is three times smaller than the shadow.

If I measure the length of the whole shadow and divide it by three, I find the heightof the tower.

You are not allowed to nail anything to the tower.

(addressing Micha) Is the calculation right?

Yes, it is right. But it is a historic building.

Saskia.

You can measure your own shadow. It may be half your own height. Then youmeasure the length of the shadow of the tower and you divide it by two. Then youknow the height of the tower, I think.

(addressing the whole class) Saskia says, I can measure my own shadow, or have itmeasured. I know how tall I am, my height makes a shadow this long. Supposethat my shadow is two times my height, the s;me applies to the tower. So, then Ican calculate the height of the tower.

486

(18) Bart:

(19) Saskia:

(20) Teacher:

(21) Micha:

(22) Teacher:

(23) Pupils:

(24) Teacher:

E. ELBERS & L. STREEFLAND

Coming back to the strip of wood. You nail the strip to the tower at a height of twometers. Then, just wait till the shadow of the strip is also two meters in the shadowof the tower. At that moment you can measure the height of the tower.

Then you have to wait a long time.

Then you have the exact height of the tower. Peter says: I have to make acalculation. The shadow gives six meters, while in reality it is two. So Richardsays the real tower has to be three times smaller as well. Only there is morecalculation to do than in Bart's solution.

But the tower of Pisa is leaning, so the shadow is much shorter. You never know ifa tower is leaning a bit.

We have to assume that the tower stands straight up from the ground.

But it isn't the tower of Pisa.

That's true, but Micha wants to indicate that our assumptions have to be veryprecise, because if the tower is leaning, the shadow is different.

At least three cycles of argumentation can be observed in this class discussion. A firstpresentation of an argument is later repeated, or rather reconstructed, twice.

To begin with, Bart introduces the idea that at a certain time of the day the shadow equalsthe tower (1-4). However, he does not succeed in transmitting this idea to the whole classroom.

Then (first argumentation cycle, statements 5-9), Peter very creatively suggests that we canattach a strip of wood to the tower at a certain distance from the base of the tower and measure theprojection of this distance in the shadow. Therefore, we can calculate the "scale of the shadow".

This discussion is then repeated by Saskia, who introduces the idea that it is notnecessary to use a strip of wood: your own shadow can be used to calculate the scale of theshadow (16, second argumentation cycle).

Then (18, third argumentation cycle), Bart reintroduces his idea. Now he is able to usethe common knowledge which the class has attained since his first contribution (in lines 1-4),by pointing to a specific case: if the shadow of the strip exactly equals the real distance thenwe know that the height of the tower is the same as the length of the shadow.

The three cycles follow a statement by Bart that the classroom did not understand (1-4).Bart argued in (I) and (3) that the shadow equals the tower when the sun is in a certainposition. At first, Bart could not put this statement across in a convincing way and the teacherdid not ask everybody's attention for Bart's idea. Later Bart's idea was accepted (18ff.),because he could reformulate his idea by making use of the statements made by Peter andSaskia about finding out the scale of the shadow.

To summarize: in this class discussion there are three cycles:

I) Attach a strip of wood to the tower for calculating the scale of the shadow.

2) Use your own shadow for calculating the scale of the shadow. This is a generalization of I.

3) The strip of wood can be used to find out when the shadow equals the tower. This is aspecial case of 1.

Not only does this discussion lead to surprisingly clever results, it also shows how thestudents and their regular teacher structure the discussion by repeating the argument, applyingit to a new situation (with the implication that we can use the shadow of any object of knownproportions) and making it more general. In the end, the students realize that they can singleout one specific case: when the length of the shadow equals the tower's height.

Discussion

The question underlying the case study presented in this article was: how do studentswho work as members of a community of inquiry regulate their collaboration and what


discursive patterns do they develop for sharing and mutually appropriating knowledge? Ifideas are proposed, what discursive tools do the participants use for circulating these ideas, forimproving them and eventually sharing them in a collaborative effort?

An analysis of the discussions in the small groups as well as in the whole classroomrevealed patterns of repetition and reconstruction that foster the circulation, improvement andacceptance of knowledge in the classroom. The discussion went in cycles in which ideas wereworked out, applied to new situations and made more general. These argumentation cycleswere used as a tool that allowed students to propose ideas, repeat them, explore and evaluatethem, and in such a form that many pupils could contribute. They typically showed partialrepetition and also elaboration of the argument. Ideas proposed by some pupils were takenover and expanded by others. In this way, the cycles contributed both to the diffusion of ideasin the classroom and to the expansion of knowledge. They allowed children to participate inthe discussions and, in doing so, to appropriate what other students had already discovered.

We have presented these observations as a case study of one lesson. We do not claim thatthis discursive pattern is characteristic of every instance of collaborative learning, nor are weconcerned here with the systematic testing of the idea of cycles of argumentation. This is workthat lies ahead. Rather, we set out to argue that in order to be successful collaborators, studentshad no option but to develop new patterns of talk. As soon as the teacher announced: "we areresearchers" and encouraged the students to generate questions and to collaborate onanswering them, the more traditional forms of interaction, which rely heavily on the teacher,became insufficient. Creating these cycles, students evoked a common context for theircollaboration and succeeded in creating continuity in discourse. The argumentation cyclesserved the aim of bringing about common knowledge in the classroom.

We showed that students' arguments in the transcripts consisted of two or more cycles,which were part of a process of progressive inquiry. The occasion for engaging in a new cyclewas often the discovery that a previous argument was based on an assumption that, on closerinspection, was not tenable or could be avoided. This is well illustrated in Sequence 4 inwhich Micha twice attempted to destroy the conclusion of the discussion. First he attacked thepossibility of attaching a strip of wood to the tower: it is forbidden to do such a thing because"it is a historic building". The teacher responded to Micha's objection by making it clear-thatthis does not refute the principle of the solution (13). Saskia's contribution in Sequence 4might also well be a reaction to Micha's opposition. She argued in (16) that it is not necessaryto attach anything to the tower, because one can use any object of known height forcalculating the scale of the shadow, including one's own shadow. So, Micha's objection led toa more general solution to the problem. At the end of the discussion he ventured a newobjection to an unwarranted assumption: the tower might be leaning, just like the tower ofPisa, and therefore the conclusion about the relationship between shadow and tower is invalid.The teacher only reacted to this statement by saying that we have to assume that the towerstands straight up from the ground. In this case the teacher chose not to continue with adiscussion about the effect of a leaning tower on the length of a shadow.

The above example shows that these children are aware that their reasoning is based onassumptions and that the validity of their arguments is dependent on the plausibility of theseassumptions. Making assumptions explicit and attacking them if necessary clearly fits intotheir roles as members of a community of inquiry. This distinguishes this form of collaborativelearning from more traditional classroom settings where students, as Reusser and Stebler(1997) demonstrated, often accept tasks with problematic assumptions, because they expectthat the tasks given to them by their teacher make sense. They have not learned to adopt acritical attitude nor to engage in argumentation in order to test the soundness of assumptions.

The teachers acted in a characteristically double role. As "teachers" they were in chargeof the classroom and decided what activities students would undertake. They introduced thetask for the lesson, and although they always chose tasks of a general nature that were suitedto collaborative work, their choice set the agenda for the classroom activities. Moreover, theyformulated the ground rules of collaborative work: they encouraged the pupils to work together,to listen to each other and to comment on others' proposed ideas. However, during the


discussions in the whole classroom, the teachers took on a different role, that of "seniorresearchers". This role brought along both opportunities and difficulties for directing the linesof argument developed by the students. Their role as senior researchers in particular allowedthem to use two discursive patterns.

First, for contributing to the construction of common knowledge in the classroom theteachers used the discursive tool of reconstructive recapitulations (Edwards & Mercer, 1987):they repeated part of what the children had said, not in a literal way, but by paraphrasing andrecasting it in a more acceptable form, eliminating errors or less preferred terminology (forinstance, line 17 in Sequence 4). However, the teacher's recaps were not used as feedback inan IRF structure, as in Edwards and Mercer's (1987) study of classroom interaction, that is, asa means for reacting to children's elicited contributions. We suggest that the teachers includedthese recaps in the argumentation cycles and used them for, as "senior researchers",participating in and contributing to the discussions.

Second, the teachers used another discursive tool. They singled out statements fordiscussion, they drew the students' attention to particular statements and made it clear thatthey were worth further exploration or discussion (for instance, line 18 in Sequence 3). Byhighlighting particular statements, the teachers influenced the appropriation of some ideasrather than others by the class.

The use of the metaphor of a community of inquiry also placed restrictions on theteachers' means for guiding the discussions in the classroom, as we showed in a differentpaper (Elbers & Streefland, 2000). The teachers were reluctant to openly make correctionsbecause of the students' status as "junior researchers". This status made it necessary for theteachers to accept the students' contributions as meriting a discussion, even if they were clearlywrong. This sometimes led to confusion. Other educational projects using a community oflearners approach have experienced comparable problems (e.g., Brown & Campione, 1994;see our discussion in Elbers & Streefland, 2000).

These constraints on the teachers arise from their new identities as "senior researchers".Only from considering the teachers' and students' identities as members of a community ofinquiry can we understand the classroom discourse and the participants' contributions to it(Elbers & Streefland, 2000). The teachers' introduction of the new roles ("from now on we areresearchers") created 11 context in which the students had to develop and use new patterns fortheir conversation and collaboration. Nobody told these children to cast their discussion intoargumentation cycles. They engaged in these cycles in order to work in accordance with theirnew identities as researchers. As the series of experimental lessons spanned only a period ofseveral months and was a separate event in an otherwise traditional school, the studentsdemonstrated considerable efforts for making the transition from their familiar learningcontext to the new situation. There was much identity talk in this classroom, especially duringthe work in groups. The children talked about the new responsibilities they had to fulfil asmembers of a community of inquiry. They also discussed their relationships with each otherand with the teachers and tried to find terms and expressions for making their cooperationpossible. To quote just one example: in the new situation, argumentation and intellectualcollaboration were more important than the solution to a problem. Even the teacher had tocome up with good arguments and justify statements. We observed several instances ofstudents not listening to or even rejecting a teacher's suggestions, because they wanted to findout an answer to a question for themselves.

The character of what counts as knowledge changes under circumstances of a communityof inquiry. The aim of the lessons was not the individual reproduction of knowledge, but thecollaborative construction of knowledge by listening to others, asking others for clarification,finding convincing arguments, formulating criticism in a constructive way, etc. Collaboratingin their role as "researchers", students learn to realize that knowledge is not something fixedor ready-made, but always needs further elaboration and is dependent on the perspectivetaken. Moreover, they come to recognize that arguing and discussing assumptions are essentialparts of knowledge and learning. In placing value on discussions and good arguments, thisclassroom resembles scientific research communities which, according to the studies by


Latour and Woolgar (1979), are characterized not only by the creative generation of new ideas,but also by participants' efforts to convince others of the value and plausibility of their ideas.

As our observations of communal learning demonstrated, ideas are not simply created ordiscovered at a particular time by particular children and then adopted or accepted by otherchildren. These ideas have to be reinvented, repeated and reconstructed by these other children.In this process, the original formulation of the ideas is often changed or amended. We havesuggested that the adoption and circulation of knowledge is found in repetitions andreconstructions of these ideas, that is: in cycles of argumentation to which many childrencontribute. Their participation in these cycles allows students to work on changing andimproving these ideas and at the same time to appropriate them as part of common knowledgein the classroom.

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490 E. ELBERS & L. STREEF LAN D

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Key wo rds : C o ll aborati ve lea rn ing , Commo n kn ow led ge , Commu nit y o f lea rne rs ,Mathem atics educa tion.

Received: June 2000

Ed Elbers, Profess or of Socia l S cience s a t U trec ht U niv ers ity . F ac u lty o f S o ci al Sc ie nc es . U trec h tU nive rs ity . P .O . B o x 80140. 3508 TC Ut rech t. The Netherl ands.

Current theme of research:

Colla borative learning in mult icultural classrooms.

MOSI releva nt publications ill lite fi eld 0/Psychology 0/ Education:

Elbers, E.. Maier. R.• Hoekstra. T.. & Hoogsreder, M. (1992). Internalization and adult -child interaction. Learning andInstruct ion , 2,1 0 1- 118.

Elbers, E.. & Kelderman , A. ( 1995). Ground rules for testing. Expectations and misund erstandings in test situations.

EuYOpell1lJourna! of Psycho!ogy of Education, 9. \ 11- \20.

Hoogsteder, M.. Maier. R.. & Elbers. E. (1996). The architec ture of adult -child interaction. Joint problem solving andthe structure o f coo peratio n. Learning and lnstrnction, 6, 345-358 .

Elbers, E., & Streefland, L. (2000 ). Shall we be researchers again '! Identit y and socia l interaction in a community ofinquiry. In H. Cowie, & D. van der Aa lsvoort (Eds.), Social inte ract ion ill learning and instruction. The meaningofdiscourse f or the construc tion of know ledge (pp. 35-5 1). Amsterdam: Pergamo n Press.

Leen Str eelland t . Unt il hi s un t ime ly de at h in 1998. Le e n Streefl a nd w a s a se nio r le c ture r a t theFreude n tha l Ins t itu te fo r Mathe m a t ic s Ed ucatio n o f Utrech t U ni ve rs it y . T he addres s o f th eFre uden thal In stitute is Tiberd reef 4. 3561 GG Utrecht. T he Ne the rlands .

Most relevan t publications inthe fi eld 0/ Psychology of Education:

Streefland, L. ( 1991). Fractions ill rea listic math ematics educatio n. A para digm of devetopmcntal research. Dordrecht:Kluwer Academic Publishers.

Streefland, L. (1993). The desig n of a mathematics course, A theoret ical reflec tion. Educat ional Snulirs ill Mathematics.

25. 109-135 .

Streetla nd. L. ( 1997). Charming fraction s or fractio ns being charmed'! In T. Nunes & P. Bryant (Eds .). Learn ing and

leaching mathematics : All in ternational perspective (pp, 3~7-37 1 ) . Hove: Psychology Press.

Elbers, E.. & Streetl and, L. (2000). Shall we be researchers again" Ident ity and soc ial interact ion in a com munity ofinqu iry. In H. Co wie & D. van der Aalsvoort (Eds.), Social interaction ;11 learning and instruc tion. The mea ning

ofdiscourse /01' the constru ction 0/ knowl etlg« (pp. 35-5 1). Amsterdam: Pergamon Press.

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Collaborative learning and the construction of common knowledge